Question

Prove the following identities :
(i) $(\sin \theta +\csc \theta {)}^2={\sin }^2\theta +{\csc }^2\theta +2$

(ii) ${\cos }^4\theta -{\sin }^4\theta =1-2{\sin }^2\theta$

Answer 1

$(\sin \theta +\csc \theta {)}^2={\sin }^2\theta +{\csc }^2\theta +2$ (ii) ${\cos }^4\theta -{\sin }^4\theta =1-2{\sin }^2\theta .$

${(sin\theta +csc\theta )}^2={sin}^2\theta +{csc}^2\theta +2sin\theta csc\theta$ ${cos}^4\theta -{sin}^4\theta =({cos}^2\theta -{sin}^2\theta )({cos}^2\theta +{sin}^2\theta )$

$={sin}^2\theta +{csc}^2\theta +2sin\theta \frac{1}{sin\theta }$ $={cos}^2\theta \left(1\right)$

$={sin}^2\theta +{csc}^2\theta +2$ $=1-2{sin}^2\theta$